About the sequence keZ we have the following a x 1 B x b for each te to b t b MKk k 1 1 x 1 te t b t b k20 c x keZ is uniformly Cauchy on t b to b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8B x b which is given by XI lim x 1 11 b to b b Thus x t lim x 1 2 x lim ff s x s ds x f S Finally the validity of this integral equation has the following two implications X 1 x f s x s ds x 0 xo dx dt f s x s ds x f f s lim x s ds x f s x s ds Thus the function tx 1 satisfying the integral equation x 1 x f s x s ds for all t 16 b 1 b aff 0 lim f s x s ds ff s x s ds f 1 X 1 This now proves that the curve X 10 8 1 82 thus obtained is a solution of the initial value problem